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Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have at least one letter repeated is
- a)69760
- b)30240
- c)99784
- d)none of these
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
Ten different letters of an alphabet are given. Words with five letter...
The total number of words that can be formed is 105 and number of these words in which no letters are repeated is 10P5.
Hence the required number
= 105 − 10P5
= 100000 − 10 × 9 × 8 × 7 × 6
= 69760
Hence the required number
= 105 − 10P5
= 100000 − 10 × 9 × 8 × 7 × 6
= 69760
This question is part of UPSC exam. View all JEE courses
Most Upvoted Answer
Ten different letters of an alphabet are given. Words with five letter...
Explanation:
Given Information:
- Total number of letters given = 10
- Words to be formed have 5 letters
Approach:
To find the number of words with at least one letter repeated, we can use the principle of inclusion-exclusion. We will first calculate the total number of 5-letter words that can be formed using the given 10 letters, and then subtract the number of words with no repeated letters, two repeated letters, three repeated letters, and four repeated letters.
Calculations:
- Total number of 5-letter words = 10P5 = 10! / (10-5)! = 10 x 9 x 8 x 7 x 6 = 30240
- Number of words with no repeated letters = 10 x 9 x 8 x 7 x 6 = 30240
- Number of words with exactly two repeated letters = 10C2 x 8 x 7 = 30240
- Number of words with exactly three repeated letters = 10C3 x 7 = 8400
- Number of words with exactly four repeated letters = 10C4 = 210
Now, using the principle of inclusion-exclusion:
Number of words with at least one letter repeated = Total - (No repeated letters + 2 repeated letters - 3 repeated letters + 4 repeated letters)
Number of words with at least one letter repeated = 30240 - (30240 - 30240 + 8400 - 210) = 69760
Therefore, the correct answer is option (a) 69760.
Given Information:
- Total number of letters given = 10
- Words to be formed have 5 letters
Approach:
To find the number of words with at least one letter repeated, we can use the principle of inclusion-exclusion. We will first calculate the total number of 5-letter words that can be formed using the given 10 letters, and then subtract the number of words with no repeated letters, two repeated letters, three repeated letters, and four repeated letters.
Calculations:
- Total number of 5-letter words = 10P5 = 10! / (10-5)! = 10 x 9 x 8 x 7 x 6 = 30240
- Number of words with no repeated letters = 10 x 9 x 8 x 7 x 6 = 30240
- Number of words with exactly two repeated letters = 10C2 x 8 x 7 = 30240
- Number of words with exactly three repeated letters = 10C3 x 7 = 8400
- Number of words with exactly four repeated letters = 10C4 = 210
Now, using the principle of inclusion-exclusion:
Number of words with at least one letter repeated = Total - (No repeated letters + 2 repeated letters - 3 repeated letters + 4 repeated letters)
Number of words with at least one letter repeated = 30240 - (30240 - 30240 + 8400 - 210) = 69760
Therefore, the correct answer is option (a) 69760.
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